KOtheory

KO-theory, or real K-theory, is a generalized cohomology theory in topology that assigns to each topological space X a graded abelian group KO^n(X). It is built from real vector bundles and stability relations, and extends the idea of complex K-theory to real vector bundles. KO-theory is represented by the KO-spectrum, and its groups are 8-periodic: KO^n(X) ≅ KO^{n+8}(X). The 8-periodicity is a manifestation of Bott periodicity in the real setting.

For a point, the coefficient groups KO^n(pt) form the eight-term sequence: KO^0(pt) ≅ Z, KO^1(pt) ≅ Z/2, KO^2(pt) ≅ Z/2, KO^3(pt) ≅ 0, KO^4(pt) ≅ Z, KO^5(pt) ≅ 0, KO^6(pt) ≅ 0, KO^7(pt) ≅ 0, repeating every eight degrees. These groups determine the “bare” information carried by KO-theory and influence computations for spaces X via spectral sequences and long exact sequences.

There are natural maps connecting KO-theory with complex K-theory KU: complexification c: KO^n(X) → KU^n(X) and realification r: KU^n(X) → KO^n(X). These maps encode how real vector bundles relate to complex ones and help transfer computations between theories.

KO-theory plays a role in index theory, obstruction theory, and the study of real vector bundles on manifolds. It provides a framework for classifying real vector bundles up to stable equivalence and for formulating invariants that detect real, as opposed to complex, structure.